There are a lot of research going on today, focusing on the search engine, analytics, Big Data processing, natural language processing, economy forecasting, dealing with reliability and certainty, medical diagnosis, pattern recognition, object recognition, biometrics, security analysis, risk analysis, fraud detection, satellite image analysis, machine generated data, machine learning, training samples, and the like.
For example, see the article by Technology Review, published by MIT, “Digging deeper in search web”, Jan. 29, 2009, by Kate Greene, or search engine by GOOGLE®, MICROSOFT® (BINGO), or YAHOO®, or APPLE® SIRI, or WOLFRAM® ALPHA computational knowledge engine, or AMAZON engine, or FACEBOOK® engine, or ORACLE® database, or YANDEX® search engine in Russia, or PICASA® (GOOGLE®) web albums, or YOUTUBE® (GOOGLE®) engine, or ALIBABA (Chinese supplier connection), or SPLUNK® (for Big Data), or MICROSTRATEGY® (for business intelligence), or QUID (or KAGGLE, ZESTFINANCE, APIXIO, DATAMEER, BLUEKAI, GNIP, RETAILNEXT, or RECOMMIND) (for Big Data), or paper by Viola-Jones, Viola et al., at Conference on Computer Vision and Pattern Recognition, 2001, titled “Rapid object detection using a boosted cascade of simple features”, from Mitsubishi and Compaq research labs, or paper by Alex Pentland et al., February 2000, at Computer, IEEE, titled “Face recognition for smart environments”, or GOOGLE® official blog publication, May 16, 2012, titled “Introducing the knowledge graph: things, not strings”, or the article by Technology Review, published by MIT, “The future of search”, Jul. 16, 2007, by Kate Greene, or the article by Technology Review, published by MIT, “Microsoft searches for group advantage”, Jan. 30, 2009, by Robert Lemos, or the article by Technology Review, published by MIT, “WOLFRAM ALPHA and GOOGLE face off”, May 5, 2009, by David Talbot, or the paper by Devarakonda et al., at International Journal of Software Engineering (IJSE), Vol. 2, Issue 1, 2011, titled “Next generation search engines for information retrieval”, or paper by Nair-Hinton, titled “Implicit mixtures of restricted Boltzmann machines”, NIPS, pp. 1145-1152, 2009, or paper by Nair, V. and Hinton, G. E., titled “3-D Object recognition with deep belief nets”, published in Advances in Neural Information Processing Systems 22, (Y. Bengio, D. Schuurmans, J. lafferty, C. K. I. Williams, and A. Culotta (Eds.)), pp 1339-1347.
One of such research and recent advances is done by Prof Lotfi Zadeh, of UC Berkeley, “the Father of Fuzzy Logic”, who recently came up with the concept of Z-numbers, plus related topics and related technologies. In the following section, we discuss the Z-numbers, taught by the U.S. Pat. No. 8,311,973, by Zadeh (issued recently).
Z-Numbers:
This section about Z-numbers is obtained from the patent by Zadeh, namely, the U.S. Pat. No. 8,311,973, which addresses Z-numbers and its applications, as well as other concepts.
A Z-number is an ordered pair of fuzzy numbers, (A,B). For simplicity, in one embodiment, A and B are assumed to be trapezoidal fuzzy numbers. A Z-number is associated with a real-valued uncertain variable, X, with the first component, A, playing the role of a fuzzy restriction, R(X), on the values which X can take, written as X is A, where A is a fuzzy set. What should be noted is that, strictly speaking, the concept of a restriction has greater generality than the concept of a constraint. A probability distribution is a restriction but is not a constraint (see L. A. Zadeh, Calculus of fuzzy restrictions, In: L. A. Zadeh, K. S. Fu, K. Tanaka, and M. Shimura (Eds.), Fuzzy sets and Their Applications to Cognitive and Decision Processes, Academic Press, New York, 1975, pp. 1-39). A restriction may be viewed as a generalized constraint (see L. A. Zadeh, Generalized theory of uncertainty (GTU)—principal concepts and ideas, Computational Statistics & Data Analysis 51, (2006) 15-46). In this embodiment only, the terms restriction and constraint are used interchangeably.
The restrictionR(X):X is A, 
is referred to as a possibilistic restriction (constraint), with A playing the role of the possibility distribution of X. More specifically,R(X):X is A→Poss(X=u)μA(u)
where μA is the membership function of A, and u is a generic value of X. μA may be viewed as a constraint which is associated with R(X), meaning that μA(u) is the degree to which u satisfies the constraint.
When X is a random variable, the probability distribution of X plays the role of a probabilistic restriction on X. A probabilistic restriction is expressed as:R(X):X isp p 
where p is the probability density function of X. In this case,R(X):X isp p→Prob(u≦X≦u+du)=p(u)du 
Note. Generally, the term “restriction” applies to X is R. Occasionally, “restriction” applies to R. Context serves to disambiguate the meaning of “restriction.”
The ordered triple (X,A,B) is referred to as a Z-valuation. A Z-valuation is equivalent to an assignment statement, X is (A,B). X is an uncertain variable if A is not a singleton. In a related way, uncertain computation is a system of computation in which the objects of computation are not values of variables but restrictions on values of variables. In this embodiment/section, unless stated to the contrary, X is assumed to be a random variable. For convenience, A is referred to as a value of X, with the understanding that, strictly speaking, A is not a value of X but a restriction on the values which X can take. The second component, B, is referred to as certainty. Certainty concept is related to other concepts, such as sureness, confidence, reliability, strength of belief, probability, possibility, etc. However, there are some differences between these concepts.
In one embodiment, when X is a random variable, certainty may be equated to probability. Informally, B may be interpreted as a response to the question: How sure are you that X is A? Typically, A and B are perception-based and are described in a natural language. Example: (about 45 minutes, usually.) A collection of Z-valuations is referred to as Z-information. It should be noted that much of everyday reasoning and decision-making is based, in effect, on Z-information. For purposes of computation, when A and B are described in a natural language, the meaning of A and B is precisiated (graduated) through association with membership functions, μA and μB, respectively, FIG. 1.
The membership function of A, μA, may be elicited by asking a succession of questions of the form: To what degree does the number, a, fit your perception of A? Example: To what degree does 50 minutes fit your perception of about 45 minutes? The same applies to B. The fuzzy set, A, may be interpreted as the possibility distribution of X. The concept of a Z-number may be generalized in various ways. In particular, X may be assumed to take values in Rn, in which case A is a Cartesian product of fuzzy numbers. Simple examples of Z-valuations are:
(anticipated budget deficit, close to 2 million dollars, very likely)
(population of Spain, about 45 million, quite sure)
(degree of Robert's honesty, very high, absolutely)
(degree of Robert's honesty, high, not sure)
(travel time by car from Berkeley to San Francisco, about 30 minutes, usually)
(price of oil in the near future, significantly over 100 dollars/barrel, very likely)
It is important to note that many propositions in a natural language are expressible as Z-valuations. Example: The proposition, p,
p: Usually, it takes Robert about an hour to get home from work,
is expressible as a Z-valuation:
(Robert's travel time from office to home, about one hour, usually)
If X is a random variable, then X is A represents a fuzzy event in R, the real line. The probability of this event, p, may be expressed as (see L. A. Zadeh, Probability measures of fuzzy events, Journal of Mathematical Analysis and Applications 23 (2), (1968) 421-427.):
      p    =                  ∫        R            ⁢                                    μ            A                    ⁡                      (            u            )                          ⁢                              p            X                    ⁡                      (            u            )                          ⁢                                  ⁢                  ⅆ          u                      ,
where pX is the underlying (hidden) probability density of X. In effect, the Z-valuation (X,A,B) may be viewed as a restriction (generalized constraint) on X defined by:Prob(X is A) is B. 
What should be underscored is that in a Z-number, (A,B), the underlying probability distribution, pX, is not known. What is known is a restriction on pX which may be expressed as:
      ∫    R    ⁢                    μ        A            ⁡              (        u        )              ⁢                  p        X            ⁡              (        u        )              ⁢                  ⁢          ⅆ      u        ⁢                  ⁢    is    ⁢                  ⁢    B  
Note: In this embodiment only, the term “probability distribution” is not used in its strict technical sense.
In effect, a Z-number may be viewed as a summary of pX. It is important to note that in everyday decision-making, most decisions are based on summaries of information. Viewing a Z-number as a summary is consistent with this reality. In applications to decision analysis, a basic problem which arises relates to ranking of Z-numbers. Example: Is (approximately 100, likely) greater than (approximately 90, very likely)? Is this a meaningful question? We are going to address these questions below.
An immediate consequence of the relation between pX and B is the following. If Z=(A,B) then Z′=(A′,1−B), where A′ is the complement of A and Z′ plays the role of the complement of Z. 1−B is the antonym of B (see, e.g., E. Trillas, C. Moraga, S. Guadarrama, S. Cubillo and E. Castiñeira, Computing with Antonyms, In: M. Nikravesh, J. Kacprzyk and L. A. Zadeh (Eds.), Forging New Frontiers: Fuzzy Pioneers I, Studies in Fuzziness and Soft Computing Vol 217, Springer-Verlag, Berlin Heidelberg 2007, pp. 133-153.).
An important qualitative attribute of a Z-number is informativeness. Generally, but not always, a Z-number is informative if its value has high specificity, that is, is tightly constrained (see, for example, R. R. Yager, On measures of specificity, In: O. Kaynak, L. A. Zadeh, B. Turksen, I. J. Rudas (Eds.), Computational Intelligence: Soft Computing and Fuzzy-Neuro Integration with Applications, Springer-Verlag, Berlin, 1998, pp. 94-113.), and its certainty is high. Informativeness is a desideratum when a Z-number is a basis for a decision. It is important to know that if the informativeness of a Z-number is sufficient to serve as a basis for an intelligent decision.
The concept of a Z-number is after the concept of a fuzzy granule (see, for example, L. A. Zadeh, Fuzzy sets and information granularity, In: M. Gupta, R. Ragade, R. Yager (Eds.), Advances in Fuzzy Set Theory and Applications, North-Holland Publishing Co., Amsterdam, 1979, pp. 3-18. Also, see L. A. Zadeh, Possibility theory and soft data analysis, In: L. Cobb, R. M. Thrall (Eds.), Mathematical Frontiers of the Social and Policy Sciences, Westview Press, Boulder, Colo., 1981, pp. 69-129. Also, see L. A. Zadeh, Generalized theory of uncertainty (GTU)—principal concepts and ideas, Computational Statistics & Data Analysis 51, (2006) 15-46.). It should be noted that the concept of a Z-number is much more general than the concept of confidence interval in probability theory. There are some links between the concept of a Z-number, the concept of a fuzzy random number and the concept of a fuzzy random variable (see, e.g., J. J. Buckley, J. J. Leonard, Chapter 4: Random fuzzy numbers and vectors, In: Monte Carlo Methods in Fuzzy Optimization, Studies in Fuzziness and Soft Computing 222, Springer-Verlag, Heidelberg, Germany, 2008. Also, see A. Kaufman, M. M. Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold Company, New York, 1985. Also, see C. V. Negoita, D. A. Ralescu, Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, 1975.).
A concept which is closely related to the concept of a Z-number is the concept of a Z+-number. Basically, a Z+-number, Z is a combination of a fuzzy number, A, and a random number, R, written as an ordered pair Z+=(A,R). In this pair, A plays the same role as it does in a Z-number, and R is the probability distribution of a random number. Equivalently, R may be viewed as the underlying probability distribution of X in the Z-valuation (X,A,B). Alternatively, a Z+-number may be expressed as (A,pX) or (μA,pX), where μA is the membership function of A. A Z+-valuation is expressed as (X,A,pX) or, equivalently, as (X,μA,pX), where pX is the probability distribution (density) of X. A Z+-number is associated with what is referred to as a bimodal distribution, that is, a distribution which combines the possibility and probability distributions of X. Informally, these distributions are compatible if the centroids of μA and pX are coincident, that is,
            ∫      R        ⁢          u      ·                        p          X                ⁡                  (          u          )                    ·                          ⁢              ⅆ        u              =                    ∫        R            ⁢              u        ·                              μ            A                    ⁡                      (            u            )                          ·                                  ⁢                  ⅆ          u                                    ∫        R            ⁢                                    μ            A                    ⁡                      (            u            )                          ·                                  ⁢                  ⅆ          u                    
The scalar product of μA and pX, μA·pX, is the probability measure, PA, of A. More concretely,
            μ      A        ·          p      X        =            P      A        =                  ∫        R            ⁢                                    μ            A                    ⁡                      (            u            )                          ⁢                              p            X                    ⁡                      (            u            )                          ⁢                                  ⁢                  ⅆ          u                    It is this relation that links the concept of a Z-number to that of a Z+-number. More concretely,Z(A,B)=Z+(A,μA·pX is B)
What should be underscored is that in the case of a Z-number what is known is not pX but a restriction on pX expressed as: μLA·pX is B. By definition, a Z+-number carries more information than a Z-number. This is the reason why it is labeled a Z+-number. Computation with Z+-numbers is a portal to computation with Z-numbers.
The concept of a bimodal distribution is of interest in its own right. Let X be a real-valued variable taking values in U. For our purposes, it is convenient to assume that U is a finite set, U={u1, . . . , un}. We can associate with X a possibility distribution, μ, and a probability distribution, p, expressed as:μ=μ1/u1+ . . . +μn/un p=p1\u1+ . . . +pn\un 
in which μi/ui means that μi, i=1, . . . n, is the possibility that X=ui. Similarly, pi\ui means that pi is the probability that X=ui.
The possibility distribution, μ, may be combined with the probability distribution, p, through what is referred to as confluence. More concretely,μ:p=(μ1,p1)/u1+ . . . +(μn,pn)/un 
As was noted earlier, the scalar product, expressed as μ·p, is the probability measure of A. In terms of the bimodal distribution, the Z+-valuation and the Z-valuation associated with X may be expressed as:(X,A,pX)(X,A,B),μA·pX is B, 
respectively, with the understanding that B is a possibilistic restriction on μA·pX.
Both Z and Z+ may be viewed as restrictions on the values which X may take, written as: X is Z and X is Z+, respectively. Viewing Z and Z+ as restrictions on X adds important concepts to representation of information and characterization of dependencies. In this connection, what should be noted is that the concept of a fuzzy if-then rule plays a pivotal role in most applications of fuzzy logic. What follows is a very brief discussion of what are referred to as Z-rules—if-then rules in which the antecedents and/or consequents involve Z-numbers or Z+-numbers.
A basic fuzzy if-then rule may be expressed as: if X is A then Y is B, where A and B are fuzzy numbers. The meaning of such a rule is defined as:if X is A then Y is B→(X,Y) is A×B 
where A×B is the Cartesian product of A and B. It is convenient to express a generalization of the basic if-then rule to Z-numbers in terms of Z-valuations. More concretely,if (X,AX,BX) then (Y,AY,BY)